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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 69696.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
69696.cd1 | 69696gq4 | \([0, 0, 0, -3068076, -2068459184]\) | \(37736227588/33\) | \(2793042278940672\) | \([2]\) | \(1474560\) | \(2.2641\) | |
69696.cd2 | 69696gq3 | \([0, 0, 0, -454476, 72811024]\) | \(122657188/43923\) | \(3717539273270034432\) | \([2]\) | \(1474560\) | \(2.2641\) | |
69696.cd3 | 69696gq2 | \([0, 0, 0, -193116, -31837520]\) | \(37642192/1089\) | \(23042598801260544\) | \([2, 2]\) | \(737280\) | \(1.9176\) | |
69696.cd4 | 69696gq1 | \([0, 0, 0, 2904, -1650440]\) | \(2048/891\) | \(-1178314711428096\) | \([2]\) | \(368640\) | \(1.5710\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 69696.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 69696.cd do not have complex multiplication.Modular form 69696.2.a.cd
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.